#math trivia #257 solution

#math trivia for #September14: #257 is a prime of the form 2^(2^n)+1. What is n? Which other day-numbers (1-365) have this form?— Burt Kaliski Jr. (@modulomathy) September 15, 2013

Answer:

n = 3 gives 2^(2^n) + 1 = 2^(2^3) + 1 = 2^8 + 1 = 257.

Other day-numbers of this form are

3 = 2^(2^0) + 1, for n = 0

5 = 2^(2^1) + 1, for n = 1

17 = 2^(2^2) + 1, for n = 2

These are all Fermat numbers, named for the mathematician who studied numbers of this form. They’re all also Fermat primes, as they both have the form and are prime numbers.

It was once conjectured that all Fermat numbers are primes, but at this point the only Fermat numbers known to be prime are the four mentioned here and the Fermat number for n = 4, which is 65537. The larger ones that have been studied are all composite — but because they grow so quickly as a function of n, it will take a long time with current methods to determine the primality of even larger ones, and nothing has yet been proved about whether other primes do or don’t exist.